ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
Determining the Optimized Initial Angle and
Velocity for Free Kicks in Soccer
Pavan Chitta
Center for Fundamental Research and Creative Education
Abstract: A direct Free Kick in soccer exemplifies the physics of projectile motion. This study investigated the
effect of projection angle and initial velocity in achieving soccer free kick goal success rates. At a fundamental
level, the initial angle, θ, and the initial velocity, V0, with which the ball is kicked, determines the trajectory of the
soccer ball and the time it takes to reach the goal. Using equations of projectile motion, as well as optimizing a
modeled equation for reaching the optimal location within optimal time, various initial values of optimized θ and
V0 have been obtained for specific Free Kick distances with the appropriate constraints.
Keywords: free-kick, football, goal, minimization, optimization, projectile, soccer.
I. INTRODUCTION
Soccer, or more widely termed as Football, has long been a cherished athletic activity in many countries across the world.
First established in 1863 as a recreational pursuit, the sport has since developed into a global competitive phenomenon,
with 209 countries fighting for the 32 coveted spots of the World Cup Finale every four years. [3] As the competition
grows, players strive to perfect their game, utilizing advanced technology to improve their fitness, speed and technical
skills. While training helps in executing offensive and defensive “plays” to some extent, these "plays” can never be
exactly the same as practised in the training due to many variables in a real match scenario. That being said, one of the
few areas of matches that can be exactly replicated is the Free Kick. A direct Free Kick is awarded when a player commits
a deliberate foul outside of his or her own penalty area, and the ensuing kick is taken from where the foul was committed.
[4]
For this project, only direct free kicks are considered, wherein the kicker can score directly off the Free Kick.
II. PROJECTILE MOTION
A projectile is an object upon which the only force acting is gravity. Gravity acts to influence the vertical motion of
the projectile, thus causing a vertical acceleration. The horizontal motion of the projectile is the result of the tendency of
any object in motion to remain in motion at constant velocity. Projectile motion is considered when a projectile is thrown
near the earth's surface, and moves along a curved path under the action of gravity only. Ideally, a football can be
considered as a projectile and its path during a kick can be traced using the equations of projectile motion, which are as
follows:
( )
................................................(1)
( )
................................................(2)
Where θ and V0 are the initial angle and initial velocity with which the ball is kicked. Gravity is represented by g. Using
both the aforementioned equations and substituting for time „t‟, we get the following equation:
( )
(
)
................................................(3)
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Paper Publications
ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
Here „y‟ is the vertical position of the ball and „x‟ is the horizontal position of the ball during its time of flight. The path
of a projectile can generally be represented by the following graph, where V x and Vy are the horizontal and vertical
velocity of the ball and Vo and θo are the initial velocity and angle of the projectile and g is the gravity.
Fig.1. Projectile motion dynamics [2]
III. PARAMETERS AND CONSTRAINTS
In the case of a soccer Free Kick, the details of the situation can be represented by the following graph, which indicates
the relevant distances and heights present. In soccer, the standard height of a goal post in an 11-a-side match is 2.44
metres. Also, the defender wall must stand a distance of 9.1 metres away from the ball during a Free Kick. The average
height of a defender is 1.9 metres and that has been taken as the height of the defender wall. The specific distance that is
considered for this project is 20 metres.
Fig. 2. Soccer Free Kick - projectile motion with constraints
Using the equations of projectile motion, data was obtained to mark the path of the football. Appropriate range constraints
were applied to both the initial angle θ and the initial velocity V 0:
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Paper Publications
ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
Also, a constraint has to be applied to filter out paths that are not able to clear the initial wall of 1.9m by the non-linear
constraint derived from equation (3) above:
( )
(
)
(Here 9.1 is the horizontal distance(x) and 1.9m is the height of the wall) Using the above ranges for Initial angle (15 to
70 degrees) and initial Velocity (15 m/s to 40 m/s), the position (Y) of the ball can be determine when it reaches the goal
(i.e. at X=20 meters).
IV. DATA ANALYSIS
The following table shows a sample of a larger table of data obtained for various angles (θ) and initial Velocity (V o)
calculating the corresponding position of the ball when it reaches the goal, while clearing the wall height.
Table 1 - Data calculations for free-kick projectile motion
Velocity
15
15
15
15
16
16
16
16
17
17
17
17
17
18
18
18
18
18
19
19
19
19
20
20
20
20
20
……
Angle
31
32
33
34
25
26
27
28
22
23
24
25
26
21
22
23
24
25
20
21
22
23
19
20
21
22
23
….
Error
2.03890152
1.815079387
1.596698071
1.384163037
2.19489352
1.922890544
1.653427416
1.386597471
2.008558164
1.714484937
1.421820831
1.130552549
0.84067983
1.463488222
1.156342442
0.849857331
0.543969569
0.238624463
1.069209413
0.752107474
0.435112311
0.118123023
0.79439911
0.469719539
0.144742127
0.18066228
0.506619863
…..
Ball height at the Wall
3.013319653
3.178727741
3.34563554
3.514123637
2.31371
2.476278358
2.640145656
2.805407585
2.043401313
2.205696607
2.369214207
2.534055189
2.700323031
2.056250033
2.219831273
2.384696137
2.550951365
2.718706601
2.039212298
2.203523632
2.369143941
2.536183432
1.998687322
2.163321689
2.329263443
2.496624676
2.665520828
…..
Ball height at the goal
0.16109848
0.384920613
0.603301929
0.815836963
0.00510648
0.277109456
0.546572584
0.813402529
0.191441836
0.485515063
0.778179169
1.069447451
1.35932017
0.736511778
1.043657558
1.350142669
1.656030431
1.961375537
1.130790587
1.447892526
1.764887689
2.081876977
1.40560089
1.730280461
2.055257873
2.38066228
2.706619863
……
We are defining „Error‟ as the vertical distance between the football where it reaches the goal and 2.2m (the optimum
height at the goal) at the horizontal distance of 20m (as the free kick is taken from a distance of 20m from the goal).
2.2 m is taken as the optimal location for the ball to enter the goal, allowing for the margin of goal height - ball diameter.
If the centre of the ball passes through 2.2m, the ball should enter the goal near the top-most possible height, farthest
away from the reach of the goalkeeper.
Page | 35
Paper Publications
ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
Following graph represents the 'Error' variation with different values of initial velocity and angles calculated using
projectile equations:
Fig. 3. Error Vs Velocity and Angle plot
V.
( )
We know that
(
ERROR OPTIMIZATION
)
So, an Error equation can also be obtained by taking the absolute value of the vertical difference between the football
position at the goal and 2.2m (the optimum position) at the horizontal distance of 20m, as stated earlier.
Mathematically the error constraint can be defined as:
Error = (
( )
(
)
)
Manual Optimization attempt:
An attempt was made to optimise the error equation manually using Lagrange multipliers (1), but the result was
inconsistent.
(
( )
(
)
)
constraint non-linear inequality restricted by the wall height:
( )
first take:
(
)
(
)
*
+
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Paper Publications
ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
[
[
]
]
[
]
Here we see that the variable terms cancel out and we are left with
, which is clearly inconsistent. Since the
manual approach did not yield proper results, MATLAB was used for getting the optimized values from the error equation
with different inbuilt solvers.
Matlab provides different solvers for a variety of optimization problems. Fmincon can be used to find a minimum of a
constrained non-linear multivariable function, internally using sequential quadratic programming algorithm. The SQP
algorithm combines the objective and constraint functions into a merit function. The algorithm attempts to minimize the
merit function subject to relaxed constraints.
'patternsearch' solver uses generalized pattern search (GPS) algorithm to find a sequence of points x 0, x1 , x2 ... that
approach an optimal point.
Optimization using fmincon and patternsearch solvers used for minimizing the error where the ball gets into the goal,
using 2.2m as the optimal goal entry height:
Objective Function for the optimizer:
( )
( ( ) (
))
(
(
(
( ( )
((
)
( ( )
((
)
)) )*
)
)
Where, X(1) and X(2) are the variables velocity and angle.
Non-Linear Constraints for error minimization:
Constraint is defined by taking negative value of the height of the ball at the wall and adding it to the defined wall height,
which should be less than or equal to 0:
[
]
( ( ) (
[
( )
))
(
(
(
( ( )
((
)
( ( )
(
)
)) )*
)
]
[]
Fmincon solver returned following optimized results with minimum error:
(Velocity: 16.894, Angle: 29.324)
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Paper Publications
ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
Fig.4. fmincon solver results for Error minimization
Substituting the solution derived from above solver we see the following ball positions:
Patternsearch: (Velocity: 22.654, Angle: 16.459)
Fig.5. patternsearch solver results for Error minimization
Page | 38
Paper Publications
ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
Substituting the solution derived from above solver we see the following ball positions:
Using patternsearch solver, we were able to clear the wall and also get the perfect optimal goal in less time compared to
the fmincon results.
VII. TIME OPTIMIZATION
It is also important to note that the position with which the ball enters the net is not the only determining factor of whether
a goal will be successful. The time taken for the ball to reach the goal also makes a difference in the goal success rate.
Less time means that the goalkeeper has less time to react and therefore the chances of saving decrease. Therefore it is
important to minimise time when finding the optimal initial angle and velocity. A similar process as was done when
minimising error can be used here, except that an additional constraint for the error has to be applied to make sure the ball
still reaches the above optimal range we set. We only want to consider those paths whose error is within 0.1m. This can be
seen from the following equations.
This function minimizes on time using wall height and the goal positions as constraints and the objective function uses the
projectile motion time calculation and getting the value of time based on velocity, angle and horizontal displacement.
Objective Function for the optimizer:
( )
(( ( )
( ( )
))
(√ ( )
(
( ( )
))
(
)))
;
Where, X(1) and X(2) are the variables velocity and angle.
Non-Linear Constraints for time minimization:
In addition to the wall constraint defined in the previous optimization, we have to add another constraint to minimize the
error of the ball entry at the goal by giving it a margin of 0.1m deviation from the 2.2m ideal height for the ball to enter
the goal:
[
]
( )
[
( ( ) (
))
(
(
(
( ( )
((
)
)
( ( )
)) )*
(
)
( ( ) (
))
(
(
(
( ( )
((
)
( ( )
( (
-( ( )
)
)) )*
)
( ( )
(
)
)*)
]
[]
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Paper Publications
ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
Fmincon solver returned following results: (Velocity: 23.654, Angle: 16.459)
Fig.6. fmincon results for Time minimization
Substituting the solution derived from above solver we see the following ball positions:
patternsearch solver returned following results: (Velocity: 23.843, Angle: 16.384)
Fig.7. patternsearch results for Time minimization
Page | 40
Paper Publications
ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
Substituting the solution derived from above solver we see the following ball positions:
Both the solvers returned approximately same results of angle and velocity with respect to time while meeting the
constraints reasonably.
VIII. EMPIRICAL APPROACH
Real field data has been captured from empirical investigation done in the field. A sample of Free Kicks was taken with
different initial velocity and angles. This was mapped using Tracker software and Adidas Snapshot. The table below has
mapped the goal success rate at intervals of initial velocity and angle:
Goal Efficiency vs. initial angle and velocity
70
Goal Efficiency
60
50
40
30
20
10
0
0< V ≤ 10, 10< V ≤ 20, 20< V ≤ 30, 30< V ≤ 40, 0< V ≤ 10, θ 10< V ≤ 20,
0<θ≤25
0<θ≤25
0<θ≤25
0<θ≤25
> 25
θ > 25
Fig.8. Goal Efficiency vs. initial angle and velocity
There is difference in the values obtained theoretically Vs. empirically, because there are several factors that have not
been taken into account such as the spin of the ball, wind factors and technique of kicking. However, we can see from the
general trend that goal efficiency is maximised around the range of:
This generally coincides with the data from the earlier, theoretical approach taken.
Fig. 9. Field Data Collection
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Paper Publications
ISSN 2350-1030
International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)
Vol. 2, Issue 2, pp: (33-42), Month: October 2015 – March 2016, Available at: www.paperpublications.org
IX. CONCLUSION
From the results of the various methods of solutions, one can note the nuances in the results data values. Performed
empirically, a slight amount of deviation from expected theoretical results is noted, and this could result for several
reasons. The spin of the ball, along with the wind factors has not been considered and thus could be a future improvement
to the project. Furthermore, varying techniques of the kickers may also influence the trajectory of the ball, despite having
the same initial angle and trajectory. Thus, taking a bigger sample of empirical data with several kickers will improve the
results.
At a time when soccer players around the world are trying to „perfect‟ their game, Mathematics offers the means for
significant enhancement of Free Kick performance for all types of players. Integrated within the training system, such
mathematical results could truly revolutionise the game and thus the notion of optimizing a Free Kick should be seriously
considered.
REFERENCES
[1] Dimitri P. Bertsekas, Constrained Optimization and Lagrange multipliers method. http://www.mit.edu/~dimitrib/
Constrained-Opt.pdf
[2] physicsb-2013-14.wikispaces.com
[3] "Overview of Soccer". Encyclopædia Britannica. Archived from the original on 12 June 2008. Retrieved 4
June 2008.
[4] "When is a direct free-kick awarded?". BBC Sport. 1 September 2005. Retrieved 17 December 2009.
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